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Write the interval expressing all real numbers less than or equal to or greater than or equal to
We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at and ends at which is written as
The second interval must show all real numbers greater than or equal to which is written as However, we want to combine these two sets. We accomplish this by inserting the union symbol, between the two intervals.
Express all real numbers less than or greater than or equal to 3 in interval notation.
When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.
These properties also apply to and
Illustrate the addition property for inequalities by solving each of the following:
The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.
Illustrate the multiplication property for inequalities by solving each of the following:
As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.
Solve the inequality:
Solving this inequality is similar to solving an equation up until the last step.
The solution set is given by the interval or all real numbers less than and including 1.
Solve the inequality and write the answer using interval notation:
Solve the following inequality and write the answer in interval notation:
We begin solving in the same way we do when solving an equation.
The solution set is the interval
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